Finite Group Databasebeta

← #27157077

Subgroups — #27157077

Order 12416.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261756C211yesmaximal, characteristic, complemented
2C4×#13095520C211yesmaximal, characteristic, complemented
3#20261583C214yesmaximal, complemented
4C2×#15690772C211yesmaximal, characteristic
5C2×#13095520—21no—
6C4⋊#11715897C2212yescomplemented
7C22⋊#11715897—21no—
8#15690865—21no—
9#15690865—21no—
10C4×#11715897C2212yescomplemented
11C22×D388C411yescharacteristic
12C22×C4×D97—21no—
13#15690772—21no—
14#15690772—21no—
15#15690772C412yes—
16#15690772C2212yes—
17C2×#13095488—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20#13095520—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—24no—
26C2×D388C2×C412yes—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—44no—
32#13095488—22no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97C2311yescharacteristic
38C22×C4×C97—21no—
39C22×C4×C97C2×C411yescharacteristic
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C23×C97C22⋊C411yescharacteristic
43C22×D97—24no—
44C22×D97—22no—
45C22×D97—42no—
46C22×D97—42no—
47C2×C4×C97—22no—
48C2×C4×C97—22no—
49C2×C4×C97—42no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97—22no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C2×D411yescharacteristic
57C2×Dic97C2×D411yescharacteristic
58D388—24no—
59D388—22no—
60D388M1614yescomplemented
61C4×D97—44no—
62C4×D97—44no—
63#11715897—24no—
64#11715897—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97#10412yes—
68C22×C97C2×C22⋊C411yescharacteristic
69C2×D97—24no—
70C2×D97—48no—
71C2×D97—42no—
72C2×D97—42no—
73C4×C97—21no—
74C4×C97—42no—
75C4×C97C2×M1612yes—
76Dic97—21no—
77Dic97—21no—
78Dic97—21no—
79Dic97AGL(1,ℤ/8)12yes—
80C2×C97—22no—
81C2×C97#40711yescharacteristic
82C2×C97#41811yescharacteristic
83C2×C97C2×#10411yescharacteristic
84D97—44no—
85#1490—971nomaximal, Sylow
86C97#149011yescharacteristic, complemented, Sylow
87C2×C4×D4—971no—
88#432—971no—
89C4×M16—971no—
90D4⋊C8—974no—
91C22×D4—971no—
92C23×C4—1941no—
93C2×M16—1941no—
94C4×D4—972no—
95C4×D4—972no—
96C4×D4—1941no—
97C4×D4—1941no—
98C2×C4⋊C4#8760411yescharacteristic
99C2×C22⋊C4—1941no—
100C2×C42—971no—
101C4⋊C8—972no—
102C22⋊C8—1941no—
103#97—1941no—
104#97—1941no—
105C4×C8—972no—
106C24—1941no—
107C2×D4—972no—
108C2×D4—1944no—
109C2×D4—1941no—
110C22×C4—21no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—3884no—
114C22×C4—1942no—
115C22×C4—1942no—
116C22×C4C2×#8760411yescharacteristic
117M16—1944no—
118C2×C8—1942no—
119C2×C8—1942no—
120C4⋊C4—21no—
121C4⋊C4C2×#8760412yes—
122C22⋊C4—1942no—
123C22⋊C4—1942no—
124C42—971no—
125C42—971no—
126C42—971no—
127C42—971no—
128C23—3882no—
129C23—3882no—
130C23—1944no—
131C23—1942no—
132C23C22⋊#8760411yescharacteristic
133D4—974no—
134D4—1944no—
135D4—1942no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—971no—
140C2×C4—971no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—3884no—
144C2×C4—3884no—
145C2×C4—1942no—
146C2×C4—1942no—
147C2×C4C22⋊#8760411yescharacteristic
148C2×C4C22×#8760411yescharacteristic
149C8—3882no—
150C8—1944no—
151C22—21no—
152C22—21no—
153C22—3888no—
154C22—3882no—
155C22—3882no—
156C22—1944no—
157C22#1569087512yes—
158C22C2×C22⋊#8760411yescharacteristic
159C4—21no—
160C4—42no—
161C4—972no—
162C4—1941no—
163C4—1941no—
164C4—1941no—
165C4D97⋊M1612yes—
166C2—22no—
167C2—3884no—
168C2#2026170711yescharacteristic
169C2#2026172711yescharacteristic
170C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1490
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261756C2semidirect
2. C4×#13095520C2semidirect
3. #20261583C2semidirect
6. C4⋊#11715897C22semidirect
10. C4×#11715897C22semidirect
60. D388M16semidirect
86. C97#1490semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202617562
C4×#130955202
#202615832
C2×#156907722
#149097